Display (5.4), p. 7 (Filmus–Ward) — $\frac{e}{e-1} g(A) \ge g(B) + \sum_i [h(A) - h(A - a_i + b_i)]$

Let $X$ be a finite set, $g : 2^X \to \mathbb{R}_{\ge 0}$ monotone increasing and submodular, and $h$ the Filmus–Ward potential of $g$,

Let $A$ and $B$ be bases of a matroid $\mathcal M$ on $X$ and $\pi : A \to B$ a bijection with $A - a + \pi(a)$ a base for every $a \in A$ (as in Lemma 5.1); write $A = \{a_1, \dots, a_r\}$ and $b_i = \pi(a_i)$. Then

This is Filmus and Ward's inequality [17, Theorem 5.1]: the total gain available from the swaps $A - a_i + b_i$ in the potential $h$ controls how far $g(A)$ can fall below $(1 - e^{-1})\, g(B)$.

Formalization Note The page states (5.4) "for any submodular function $g$", citing [17], whose setting is monotone submodular $g$ normalised by $g(\emptyset) = 0$; the paper applies it only to the monotone nonnegative $g$ of Lemmas A.1 and A.2. The statement therefore assumes $g$ monotone increasing and nonnegative. The $B = \emptyset$ term of $h$ is excluded (see the Setting definition); with that convention the inequality holds for $g(\emptyset) \ge 0$, since the constant $g(\emptyset)$ adds the same amount to $h(A)$ and $h(A - a_i + b_i)$. The indexing $b_i = \pi(a_i)$ is a sum over $a \in A$.